Wednesday, October 12, 2005

Hypothesis, Theory, and Proof (very long)

One of the classes I'm taking this semester is a senior seminar entitled "Nature and Origins of Consciousness." We've looked at and discussed different historical explanations for consciouness and have now wandered into a discussion of intelligent design vs evolution. Which leads me to the concepts of hypothesis, theory, and proof (and a rant).

I'm finding that scientists, psychologists, mathematicians, and the general population use each of these terms in somewhat different ways. Everyone seems to use hypothesis to denote an unproved idea. A scientific or mathematical hypothesis appears to be a complete idea. It's the theory and proof concepts where problems occur.

For scientists and mathematicians, a theory (or theorem) is an hypothesis that has been proven. For psychologists, a theory is a logically organized set of statements that include definitions of various events or concepts, contain information about relationships between these events, explain the causes of the events where the hypothesis is simpler and more tentative. (I just had to memorize this for a test.)

Psychologists and scientists consider a theory proved when they can show significant evidence for it and when the theory exhibits good predictive validity. Einstein's theory of relativity was considered proven once phenomena predicted by the theory was shown to exist and when phenomena behaved in ways the theory predicted. In psychology proofs of theories are really statistical analyses of experimental data showing that the hypothesis based on the theory has a good probability of being true.

Since people (and the larger world in general) aren't particulary tidy, proof is not a 100% probability, true in all cases, concept for psychologists or scientists. This is also one of the reasons why theories are changed (or thrown out completely). As we learn more and gain more evidence, theories are modified. Sometimes the modifications become so unwieldy that science is forced into rethinking the whole idea. This is how science grows, but this does not automatically invalidate science.

Mathematicians (who are anal retentive) define a proof as a logical and rigourous mathematical argument. Unfortunately exhaustive evidence is not a proof. The mathematician must show that the hypothesis is true for all instances. For example, if I want to show that the sum of the numbers from 1 to n (n is any counting number greater than 1) = n(n+1)/2, just summing all the numbers from 1 to n=2, 1 to n=3, 1 to n=4, and so on for however long I want to go doesn't constitue proof. I could have done the sums a million times and all equaled n(n+1)/2, but that's not a proof. The proof involves an argument that shows this is true for all values of n. (BTW - this has been proved.)

The nice thing about this? Once a theorem is proved in mathematics, it stays proved. Euclid showed a proof of the PythagoreanTheorem that is still valid. The not so nice thing? Hypothesis that appear to work and for which there is a lot of evidence, but no proof. My personal favorite is the Riemann Hypothesis - unproven, but over a million solutions have been calculated and all agree with the hypothesis.

Remember what I said about people not being as tidy as numbers? I have to remind myself of this on a regular basis. Otherwise, I can't look at any psychological theory as proven. I learned the concept of proof in terms of mathematics.

Looking at the arguments "against" Darwin's theory of evolution, it seems that the general population defines both hypothesis and theory as an unproven idea and expects proof to be similar to a mathematicians - 100% true in all cases. The fact that evolutionary theory has significant amounts of evidence and excellent predictive value doesn't appear to convince people of its validity. I end up with a problem though when the counter proposal involves an unknown intelligence for which no evidence is available and a theory with no apparent predictive value.

Of course I also have a problem with the definition of a transcendental number - a number which is not algebraic. Hello, tell me what it is not what it isn't.


Contagion said...

I had this class in college... Thanks for the flashback. :)

MathCogIdiocy said...

Is a flashback a good thing? If so, I live to please.

nellsquirrel said...

But we LIVE to tell you what it's not... *ducking and running from the mad Math person*

MathCogIdiocy said...

Nell - you are toast.

Irishpixie said...

Uh, Jacqui, can I borrow half your brain?

MathCogIdiocy said...

You could borrow half my brain, but then I'd be totally brainless. *bg*

Harvey said...

I'm guessing this falls under the "MathCog" portion of your blog's title :-)

MathCogIdiocy said...

Harvey - this certainly is part of the "MathCog" and unfortunately probably not the last you'll see of it. As the research project takes shape and my search for a graduate program goes on, wandering thoughts on both will appear. Be afraid, be very afraid. :)